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Question

# Let A and B any two independent events with 0<P(A)<1 and 0<P(B)<1. Which of the following statements are TRUE? I. P(A occurs but B does not occur)=P(A)âˆ’P(Aâˆ©B) II. P(Exactly one of A and B occurs)=P(A)+P(B)âˆ’P(Aâˆ©B) III. P(Neither A nor B occurs)=(P(A)âˆ’1)(P(B)âˆ’1) IV. P(A occurs given that B does not occur)=P(A)

A
Only I, II, III
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B
Only I, II, IV
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C
Only I, III, IV
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D
All of them
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Solution

## The correct option is C Only I, III, IVWe know that if A and B are independent, then 1. A and BC are independent. 2. AC and B are independent. 3. AC and BC are independent. I. P(A occurs but B does not occur) =P(A∩BC) =P(A)P(BC) =P(A)(1−P(B)) =P(A)−P(A)P(B) =P(A)−P(A∩B) II. P(Exactly one of A and B occurs) =P((A∩BC) ∪ (AC∩B)) =P(A)P(BC)+P(AC)P(B) =P(A)+P(B)−2P(A∩B) III. P(Neither A nor B occurs) =P(AC∩BC) =P(AC)P(BC) =(1−P(A))(1−P(B)) =(P(A)−1)(P(B)−1) IV. P(A occurs given that B does not occur) =P(A|BC) =P(A∩BC)P(BC) =P(A)P(BC)P(BC)=P(A) Alternatively, we can directly conclude that P(A|BC)=P(A) as A and BC are independent.

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